The Generalized Lyapunov Demodulator: High-Bandwidth, Low-Noise Amplitude and Phase Estimation

Effective demodulation of amplitude and phase is a requirement in a wide array of applications. Recent efforts have increased the demodulation performance, in particular, the Lyapunov demodulator allows bandwidths up to the carrier frequency of the signal. However, being inherently restricted to first-order filtering of the input signal, it is highly sensitive to frequency components outside its passband region. This makes it unsuitable for certain applications such as multifrequency atomic force microscopy (AFM). In this article, the structure of the Lyapunov demodulator is transformed to an equivalent form and generalized by exploiting the internal model principle. The resulting generalized Lyapunov demodulator structure allows for arbitrary filtering order and is easy to implement, requiring only a bandpass filter, a single integrator, and two nonlinear transformations. The generalized Lyapunov demodulator is implemented experimentally on a field-programmable gate array (FPGA). Then it is used for imaging in an AFM and benchmarked against the standard Lyapunov demodulator and the widely used lock-in amplifier. The lock-in amplifier achieves great noise attenuation capabilities and off-mode rejection at low bandwidths, whereas the standard Lyapunov demodulator is shown to be effective at high bandwidths. We demonstrate that the proposed demodulator combines the best from the two state-of-the-art demodulators, demonstrating high bandwidths, large off-mode rejection, and excellent noise attenuation simultaneously.